Finite Group Databasebeta

← #27278594

Subgroups — #27278594

Order 13568.

412 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283782C211yesmaximal, characteristic
2#20273506C212yesmaximal, complemented
3#20273377C211yesmaximal, characteristic
4#20272548C212yesmaximal, complemented
5C53×#1962C211yesmaximal, characteristic
6#16810701—22no—
7D4×#88322—22no—
8#16810676C2212yes—
9#16810676C2212yes—
10#16810590—22no—
11#16810375—22no—
12#16810373—22no—
13#16810372C412yescomplemented
14#16810367C411yescharacteristic, complemented
15#16810367C411yescharacteristic, complemented
16C53×#532C2211yescharacteristic
17C53×#423C2211yescharacteristic
18C2×#104×C53—22no—
19C43×C53C2211yescharacteristic
20C2×#11719917—22no—
21C22×#88322—22no—
22C4⋊#88322C2×C412yes—
23C4⋊#88322C2×C412yes—
24C4⋊#88322D412yes—
25C4⋊#88322D412yes—
26C22⋊#88322—24no—
27C22⋊#88322—22no—
28C22⋊#88322—22no—
29C4×#88322—22no—
30C4×#88322—22no—
31C2×#41×C53D412yes—
32C2×M16×C53—21no—
33C4⋊Q8×C53C2×C411yescharacteristic
34#127×C53C2×C411yescharacteristic
35#124×C53Q812yes—
36#121×C53—22no—
37C4×D4×C53—22no—
38C2×C42×C53—21no—
39C2×C42×C53—22no—
40C2×C42×C53C2311yescharacteristic
41C4⋊C8×C53—22no—
42#104×C53—44no—
43#41×C53—42no—
44#41×C53—42no—
45C2×Q8×C53C4⋊C412yes—
46C2×D4×C53—22no—
47C2×D4×C53—22no—
48C2×D4×C53C4⋊C412yes—
49C22×C4×C53—22no—
50C22×C4×C53—41no—
51C22×C4×C53C2×D411yescharacteristic
52C22×C4×C53#4112yes—
53M16×C53—42no—
54C2×C8×C53—41no—
55C4⋊C4×C53—22no—
56C22⋊C4×C53—24no—
57C42×C53—22no—
58C42×C53—24no—
59C42×C53—42no—
60C42×C53—41no—
61C42×C53C22×C411yescharacteristic
62C42×C53C2×D412yes—
63C42×C53C2×Q811yescharacteristic
64#11719917—82no—
65C2×#88322—22no—
66C2×#88322—22no—
67C2×#88322—22no—
68C2×#88322—22no—
69C2×#88322—82no—
70C23×C53—22no—
71C23×C53#12311yescharacteristic
72Q8×C53—42no—
73D4×C53—42no—
74D4×C53—42no—
75D4×C53—42no—
76D4×C53—82no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—22no—
82C2×C4×C53—42no—
83C2×C4×C53—44no—
84C2×C4×C53—42no—
85C2×C4×C53—42no—
86C2×C4×C53C2×C4⋊C411yescharacteristic
87C2×C4×C53C4×D412yes—
88C2×C4×C53#12011yescharacteristic
89C2×C4×C53C22⋊Q812yes—
90C8×C53—81no—
91#88322—42no—
92#88322—42no—
93#88322—42no—
94#88322—42no—
95#8934—531nomaximal, Sylow
96C22×C53—21no—
97C22×C53—22no—
98C22×C53—42no—
99C22×C53—44no—
100C22×C53#38511yescharacteristic
101C4×C53—21no—
102C4×C53—22no—
103C4×C53—44no—
104C4×C53—42no—
105C4×C53#44312yes—
106C4×C53#44412yes—
107C4×C53#45311yescharacteristic
108C4×C53#45511yescharacteristic
109#2979—531no—
110#1962D5311yescharacteristic
111#1574—532no—
112#1543—531no—
113#1276—532no—
114C2×C53—41no—
115C2×C53—82no—
116C2×C53#185611yescharacteristic
117C2×C53#186612yes—
118#532C2×D5311yescharacteristic
119#436—1062no—
120C8×D4—1062no—
121#431—532no—
122#431—532no—
123#423C2×D5311yescharacteristic
124C2×#104—22no—
125#408—1062no—
126C43C2×D5311yescharacteristic
127#335—1062no—
128#334—1062no—
129#333—532no—
130#331—531no—
131#331—531no—
132C53#893411yescharacteristic, complemented, Sylow
133C2×#41C22⋊D5312yes—
134C2×M16—21no—
135C2×M16—1062no—
136C22×C8—1062no—
137C4⋊Q8C4×D5311yescharacteristic
138#127C4×D5311yescharacteristic
139#124Dic10612yes—
140#121—22no—
141C4×D4—22no—
142C2×C42—22no—
143C2×C42—21no—
144C2×C42C22×D5311yescharacteristic
145C4⋊C8—22no—
146C4⋊C8—532no—
147C4⋊C8—532no—
148C4⋊C8—532no—
149C4⋊C8—532no—
150#104—44no—
151C22⋊C8—1064no—
152C22⋊C8—1062no—
153C22⋊C8—1062no—
154C4×C8—1062no—
155C4×C8—1062no—
156#41—42no—
157#41—42no—
158C2×Q8#1171991912yes—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4#1171991912yes—
162C22×C4—22no—
163C22×C4—41no—
164C22×C4#1171994712yes—
165C22×C4C2×C22⋊D5311yescharacteristic
166M16—42no—
167M16—4242no—
168C2×C8—41no—
169C2×C8—1062no—
170C2×C8—1062no—
171C2×C8—1062no—
172C2×C8—1062no—
173C2×C8—4242no—
174C4⋊C4—22no—
175C22⋊C4—24no—
176C42—22no—
177C42—24no—
178C42—41no—
179C42—42no—
180C42C2×Dic10611yescharacteristic
181C42C2×C4×D5311yescharacteristic
182C42D4×D5312yes—
183C23—22no—
184C23#1313319811yescharacteristic
185Q8—42no—
186D4—42no—
187D4—42no—
188D4—42no—
189D4—82no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—44no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4C2×#1171991911yescharacteristic
200C2×C4#1313317712yes—
201C2×C4#1313318511yescharacteristic
202C2×C4#1313319412yes—
203C8—81no—
204C8—2122no—
205C8—2122no—
206C8—2122no—
207C8—2122no—
208C22—22no—
209C22—21no—
210C22—44no—
211C22—42no—
212C22#1681049511yescharacteristic
213C4—22no—
214C4—21no—
215C4—44no—
216C4—42no—
217C4#1681073712yes—
218C4#1681074212yes—
219C4#1681079011yescharacteristic
220C4#1681080011yescharacteristic
221C2—41no—
222C2—82no—
223C2#2027492411yescharacteristic
224C2#2027498812yes—
subgroupG/H
centerC2×C4#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1962C2
socleC22×C53#385
prime radicalC53×#532C22
cosocleC2×C42×C53C23
hypercenter#1962D53
hypocenterC53#8934
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1962

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20273506C2semidirect
4. #20272548C2semidirect
13. #16810372C4semidirect
14. #16810367C4semidirect
15. #16810367C4semidirect
132. C53#8934semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202837822
#202735062
#202733772
#202725482
C53×#19622
#893453